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151.
This paper considers the asymptotic properties of two kernel estimates % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaacamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E82!\[\tilde f_{_n }\]and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaqcamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E83!\[\hat f_{_n }\], which have been proposed by Bhattacharyya et al. (1988, Comm. Statist. Theory Methods, A17, 3629–3644) and Jones (1991, Biometrika, 78, 511–519), respectively, for estimating the underlying density f at a point under a general selection biased model. The asymptotic optimality of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaqcamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E83!\[\hat f_{_n }\]and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaacamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E82!\[\tilde f_{_n }\]is measured by the corresponding asymptotic minimax mean squared errors under a compactly supported Lipschitz continuous family of the underlying densities. It is shown that, in general, % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaqcamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E83!\[\hat f_{_n }\]is a superior local estimate than % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaacamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E82!\[\tilde f_{_n }\]in the sense that the asymptotic minimax risk of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaqcamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E83!\[\hat f_{_n }\]is lower than that of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaacamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E82!\[\tilde f_{_n }\]. The minimax kernels and bandwidths of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaqcamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E83!\[\hat f_{_n }\]are computed explicity and shown to have simple forms and depend on the weight functions of the model. 相似文献
152.
This paper presents an error analysis for classification algorithms generated by regularization schemes with polynomial kernels.
Explicit convergence rates are provided for support vector machine (SVM) soft margin classifiers. The misclassification error
can be estimated by the sum of sample error and regularization error. The main difficulty for studying algorithms with polynomial
kernels is the regularization error which involves deeply the degrees of the kernel polynomials. Here we overcome this difficulty
by bounding the reproducing kernel Hilbert space norm of Durrmeyer operators, and estimating the rate of approximation by
Durrmeyer operators in a weighted L1 space (the weight is a probability distribution). Our study shows that the regularization parameter should decrease exponentially
fast with the sample size, which is a special feature of polynomial kernels.
Dedicated to Charlie Micchelli on the occasion of his 60th birthday
Mathematics subject classifications (2000) 68T05, 62J02.
Ding-Xuan Zhou: The first author is supported partially by the Research Grants Council of Hong Kong (Project No. CityU 103704). 相似文献
153.
154.
GINDIKIN Simon 《中国科学A辑(英文版)》2006,49(11):1655-1661
We discuss some problems on rigidity of canonical realizations of complex homogeneous domains at ℂn
Dedicated to Professor Sheng GONG on the occasion of his 75th birthday 相似文献
155.
In this paper, we establish sharp two-sided estimates for the Green functions of non-symmetric diffusions with measure-valued drifts in bounded Lipschitz domains. As consequences of these estimates, we get a 3G type theorem and a conditional gauge theorem for these diffusions in bounded Lipschitz domains.Informally the Schrödinger-type operators we consider are of the form L+μ⋅∇+ν where L is a uniformly elliptic second order differential operator, μ is a vector-valued signed measure belonging to Kd,1 and ν is a signed measure belonging to Kd,2. In this paper, we establish two-sided estimates for the heat kernels of Schrödinger-type operators in bounded C1,1-domains and a scale invariant boundary Harnack principle for the positive harmonic functions with respect to Schrödinger-type operators in bounded Lipschitz domains. 相似文献
156.
Ding Qing 《数学年刊B辑(英文版)》1994,15(1):35-42
This paper establishes a new Laplacian comparison theorem which is specially useful to the manifolds of nonpositive curvature.It leads naturally to the corresponding heat kernel comparison and eigenvalue comparison theorems. Furthermore, a lower estimate of L^2-spectrum of an n-dimensional non-compact complete Cartan-Hadamard manifold is given by (n-1)k/4,provided its Ricci curvature ≤-(n-1)k(k=const.≥0). 相似文献
157.
Spatial Nonparametric Regression Estimation: Non-isotropic Case 总被引:3,自引:0,他引:3
Zu-di Lu Xing ChenInstitute of Systems Science Academy of Mathematics Systems Sciences Chinese Academy of Sciences. Beijing ChinaDepartment of Statistics Yunnan University Kunming China 《应用数学学报(英文版)》2002,18(4):641-656
Data collected on the surface of the earth often has spatial interaction. In this paper, a non-isotropic mixing spatial data process is introduced, and under such a spatial structure a nonparametric kernel method is suggested to estimate a spatial conditional regression. Under mild regularities, sufficient conditions are derived to ensure the weak consistency as well as the convergence rates for the kernel estimator. Of interest are the following: (1) All the conditions imposed on the mixing coefficient and the bandwidth are simple; (2) Differently from the time series setting, the bandwidth is found to be dependent on the dimension of the site in space as well; (3) For weak consistency, the mixing coefficient is allowed to be unsummable and the tendency of sample size to infinity may be in different manners along different direction in space; (4) However, to have an optimal convergence rate, faster decreasing rates of mixing coefficient and the tendency of sample size to infinity along each direction a 相似文献
158.
本文运用再生核Hhilbert空间方法研究R^s中响应曲面模型的稳健设计问题。我们假设模型偏差包括由多元Hermite多项式高阶项产生的效应,如果偏差大,设计点应该布于布点附近,否则,设计点应该适当散开,所求设计是正交不变的,而且关于模型偏差是稳健的。 相似文献
159.
Kernel regression estimation for continuous spatial processes 总被引:1,自引:0,他引:1
We investigate here a kernel estimate of the spatial regression function r(x) = E(Y
u | X
u = x), x ∈ ℝd, of a stationary multidimensional spatial process { Z
u = (X
u, Y
u), u ∈ ℝ
N
}. The weak and strong consistency of the estimate is shown under sufficient conditions on the mixing coefficients and the
bandwidth, when the process is observed over a rectangular domain of ℝN. Special attention is paid to achieve optimal and suroptimal strong rates of convergence. It is also shown that this suroptimal
rate is preserved by using a suitable spatial sampling scheme.
相似文献
160.
本文讨论一类与石油试井有密切联系的带有等位面边界条件的抛物型偏微分方程,对带有位面边界条件的热核函数进行了讨论,对方程解的梯度进行了估计,并给出了Harnack型不等式,一类非线性抛物型方程与热方程的比较定理,以及在试井分析中应用的一个例子。 相似文献